Had ministers let probation officers monitor offenders rather than signing contracts with private-sector firms, the money saved could have been used to recruit an extra 1,200 police officers, Policy Exchange said.
Its new report claimed the system used in England and Wales has changed little since 1989, with people serving the last few months of jail terms or community sentences forced to wear ankle bracelets linked to a box in their homes and kept under a curfew for 12 hours a night.
However it means that prolific offenders such as burglars and shoplifters can still go out at commit crimes during daylight hours, without the authorities knowing where they are.
Policy Exchange proposes that the new contract set to be signed by the Ministry of Justice and one or two private firms to run tagging schemes, worth as much as £3billion, should be torn up in favour of more competition.
The think-tank also wants GPS satellite tags to be used in future so offenders can be followed 24 hours a day.
Police could even map their movements against reported crimes, or plot their location in real-time when responding to 999 calls to see if they were at the scene.
Under the biggest proposed expansion of the scheme, which would help keep the prison population down, the numbers being tagged could rise from 25,000 to 140,000 with suspects on bail, criminals given suspended sentences and sex offenders included.
The think-tank said £963m had been spent on electronic monitoring over the past 13 years but criminal justice experts had not been allowed to help develop the service, with Serco and G4S enjoying a virtual monopoly in recent years.
It calculated that the core service works out at £13.14 a day in England and Wales, compared with £1.22 in the US.
If the American model had been followed of letting local offender management teams carry out supervision, with private firms only providing the hardware, as much as £883m could have been freed up.
Policy Exchange said its surveys suggested as many as one in four police forces think the current regime is ineffective, with some officers calling it "expensive" and "unreliable".
Rory Geoghegan, the author of the report, said: "The public and the police support tagging criminals. But we need to be confident that the public is getting the protection and the value for money they need and should rightly expect. Extending the use of tagging without these reforms will just see millions of pounds wasted and a real opportunity to cut crime missed.
"As technology becomes ever more sophisticated our badly designed procurement system is preventing the police from preventing and detecting crime. We desperately need to create a real market so that the police can get the technology they need to cut burglary, cut robbery and other crimes that have a massive impact on victims and the community."